Question: $ 0.\overline{96} \div 1.\overline{17} = {?} $
Explanation: First convert the repeating decimals to fractions. $\begin{align*} 100x &= 96.9696...\\ x &= 0.9696...\end{align*} $ $\begin{align*} 99x &= 96 \\ x &= \dfrac{96}{99}\end{align*} $ $\begin{align*} 100y &= 117.1717...\\ y &= 1.1717...\end{align*} $ $\begin{align*} 99y &= 116 \\ y &= \dfrac{116}{99}\end{align*} $ So, the problem becomes: $ \dfrac{96}{99} \div \dfrac{116}{99} = {?} $ Dividing by a fraction is the same as multiply by the reciprocal of that fraction. $ \dfrac{96}{99} \times \dfrac{99}{116} = {?} $ $ \phantom{\dfrac{96}{99} \times \dfrac{116}{99}} = \dfrac{96 \times 99}{99 \times 116} $ $ \phantom{\dfrac{96}{99} \times \dfrac{116}{99}} = \dfrac{96 \times \cancel{99}} {\cancel{99} \times 116} $ $ \phantom{\dfrac{96}{99} \times \dfrac{116}{99}} = \dfrac{96}{116} $ Simplify: ${= \dfrac{24}{29}}$